Appeal from the decision of the Patent and Trademark Office Board of Appeals (board), affirming the rejection of claims 5-7, all the claims in application serial No. 68,507, filed August 31, 1970, entitled “Method of Generating a Curve, Applicable *1238to Tracing Machines or Machine Tools.” We affirm.
The Invention
De Castelet’s invention relates to a method of generating a curve or family of curves, employing a computer in conjunction with drafting and milling machines. The invention employs these principles: (1) a curve may be represented by four points defining two straight line tangents to the ends of the curve, and (2) a curve can be represented by a transformed base-of-reference curve, by transforming coordinates of points representing a unit or base curve into a coordinate system of a curve to be generated.
In de Castelet’s overall system:
data are inputed to computer 10 through control boards 19 and 20. Calculations, using the inputed data and pursuant to stored programs, are performed by computer 10. Computer 10 also receives milling machine 11 position information from encoder 13 and drafting machine 12 position information from encoder 14. Based upon the results of its calculations and information received from encoders 13 and 14, computer 10 generates speed-change signals for controlling position motor driving shafts 21-25. The net result of the operation is a machined surface (milling machine 11) or a drawing (drafting machine 12) which corresponds to a desired curve formation.
According to de Castelet’s method, curves to be generated are first broken down into a series of successive arcs. Those arcs can be defined by two straight line tangents, for example, AM and BN in the following illustrations:
The data inputed to the computer constitutes the X, Y, and Z Cartesian coordinates of points A, M, N, and B. The computer, pursuant to a stored program, treats each arc so defined as though it were the transformed curve of a base curve inscribed within and between two opposite vertices of a unit cube. The coordinates of that base curve are transformed by computer calculations, according to the following transformation equations, into coordinates of the curve to be generated:
De Castelet claims:
7. A machine method of generating a curve from data supplied to a computer in the form of coordinates of points defining two given segments of tangents to the curve to be generated extending from the end of and subtended by said curve for controlling numerical control system type model forming means, wherein data, in the form of electrical signals representing a table of coordinates of points, of characteristics of a base curve inscribed on a unit-cube between two oppo*1239site vertices of said unit-cube is stored in a memory bank of said computer and said computer is programmed (1) to treat electrical signals representing a given arc, defined by the coordinates of the ends of two segments of tangents extending from and subtended by the ends of said given arc, as the transformed curve of said base curve, wherein said ends are considered as the transformed points of vertices of said unit-cube and (2) to calculate and transmit to the control system of said model forming means electrical signals representing the coordinates of a sequence of points of successive ones of said given arc defining said curve to be generated, said computer thereafter automatically performing the steps of:
(a) transforming the electrical signals representing said coordinates of points defining two given segments of tangents to said curve to be generated by program (1) to define a corresponding change in reference coordinates with respect to the characteristics of the stored base curve;
(b) calculating the sequence of coordinates of the current points of the transformed base curve of program (1) through the change in reference coordinates obtained from step (a); and
(c) transmitting electrical signals representing said sequence of coordinates calculated in step (b) from said computer to said model forming means by program (2).
5. The method according to claim 7, wherein step (b) further comprises calculating said sequence of coordinates of the current points according to the following equations:
Xw, Yw and Zw are the coordinates of a current point of each arc of the curve to be generated;
Xa, Ya and Za are the coordinates of the point of origin of the arc of the curve to be generated; and
x, y and z are the coordinates of a current point of the base curve, and
are the respective projections on the X, Y and Z axes of the two segments of tangents to the arc of the curve to be generated and of the segment joining the ends of said two segments opposing the point of tangency.
6. The method according to claim 7, wherein step (c) comprises the further step of interpolating between adjacent points of said sequence of points obtained by program (2) according to a calculation of the director cosines of the tangents of each one of said sequence of points.
The Board
The board, citing Gottschalk v. Benson, 409 U.S. 63, 93 S.Ct. 253, 34 L.Ed.2d 273, 175 USPQ 673 (1972), stated that “where the claimed novelty involves a formula, equation or algorithmic process and has no substantial practical application except in connection with a digital computer, a patent to such would in practical effect be a patent on the algorithm itself, and should not be granted.”
Acknowledging that independent claim 7 does not recite a specific algorithm, equation, or mathematical formula, the board nevertheless found an algorithmic process contained therein, and concluded that, because the apparatus was known, any novelty in the claims resulted from that algorithmic process. That novelty, the board said: “is of the same nature as that which is condemned in Benson.”
The board was of the opinion that patenting the present claims would, in effect, preempt the algorithmic process despite the fact that “they recite a machine environment, a particular art or a particular end use.”
*1240 The Issue
As in In re Chatfield, 545 F.2d 152, 191 USPQ 730 (Cust. & Pat.App.1976), cert. denied, 46 U.S.L.W. 3203 (Oct. 4, 1977), the sole issue before us is whether the particular claims on appeal define statutory subject matter under 35 U.S.C. § 101.1
OPINION
The Board's Interpretation of Benson
Though we agree with the board’s ultimate conclusion, and with its reference to Benson as precluding patentability of claims to a mathematical equation, we disagree with several of the board’s expressions of the applicable law. Initially, the board found:
[T]he thrust of the decision in Benson to be that computer programs or program implemented algorithms are not patentable subject matter at least until such time as the Congress acts otherwise.
That “computer programs” are not patentable is not the “thrust” of Benson. As the Court cautioned:
It is said that the decision precludes a patent for any program servicing a computer. We do not so hold. [409 U.S. at 71, 93 S.Ct. at 257, 175 USPQ at 676.]
and as the Court’s characterization of Benson in Dann v. Johnston, 425 U.S. 219, 224, 96 S.Ct. 1393, 1396, 47 L.Ed.2d 692, 189 USPQ 257, 259 (1976) confirmed:
Our limited holding * * * was that respondent’s method was not a patentable “process” as that term is defined in 35 U.S.C. § 100(b). [Citations omitted.]
Absent contrary directions, no basis exists for a moratorium on protection of inventions embodying or using computer programs. Such broad prohibition could subject meritorious statutory inventions to unabatable piracy, and could forestall invention disclosure, the hallmark of the patent system, until Congress chooses to act.
We disagree, also, with the breadth of the board’s succeeding comment:
Stated another way, where the claimed novelty involves a formula, equation or algorithmic process and has no substantial practical application except in connection with a digital computer, a patent to such would in practical effect be a patent on the algorithm itself, and should not be granted.
That second comment is not a restatement of the board’s preceding comment, insofar as it envisions the question of statutory subject matter as residing in a “claimed novelty” and its practical application. As a majority indicated in Chatfield, supra, claim dissection and comparison against prior art to determine a limited “claimed novelty” is not proper. What an applicant chooses to encompass by the whole of his claims comprises his “claimed novelty.” There is no 35 U.S.C. § 103 rejection before us, based on the view that it would have been obvious to employ de Castelet’s equations in apparatus described by the examiner as old.2
The board’s focus on practical applications for the “claimed novelty” springs from the “nutshell” holding in Benson:
It is conceded that one may not patent an idea. But in practical effect that *1241would be the result if the formula for converting BCD numerals to pure binary numerals were patented in this case. The mathematical formula involved here has no substantial practical application except in connection with a digital computer, which means that if the judgment below is affirmed, the patent would wholly pre-empt the mathematical formula and in practical effect would be a patent on the algorithm itself. [409 U.S. at 71-72, 93 S.Ct. at 257, 175 USPQ at 676.]
That language is interpreted by the solicitor and the board as foreclosing patentability of the appealed claims because, as de Castelet admitted at oral argument, the only practical application of the involved algorithm (equations) is in the claimed method. But the notion that the “nutshell” language makes a method nonstatutory whenever it involves mathematical equations having their only practical use in the method is neither impelled by the rationale of Benson nor supported by precedent.
The “meat” within the “nutshell” is not that patentability is precluded whenever an inventor discloses and claims only one use for his formula. The Supreme Court felt that “Benson’s claims would have preempted all practical use of both the underlying mathematical formula and the involved algorithm,” [Emphasis added.] Chatfield, 545 F.2d at 156, 191 USPQ at 733, i. e., Benson’s claims were considered as in effect drawn to the algorithm itself.
This court, in In re Benson, 441 F.2d 682, 58 CCPA 1134, 169 USPQ 548 (1971), recognized that the only practical use of Benson’s method was in conjunction with a computer, but stated why that fact would not support a “mental steps” rejection:
Realistically, the process of claim 13 has no practical use other than the more effective operation and utilization of a machine known as a digital computer. It seems beyond question that the machines — the computers — are in the technological field, are a part of one of our best-known technologies, and are in the “useful arts” rather than the “liberal arts” as are all other types of “business machines,” regardless of the uses to which their users may put them. How can it be said that a process having no practical value other than enhancing the internal operation of those machines is not likewise in the technological or useful arts? [Emphasis in original. 441 F.2d at 688, 58 CCPA at 1143-44, 169 USPQ at 553.]
Benson’s limitation of his claims to a computer machine environment overcame the rejection bottomed on a “mental steps” theory. Because operation of a formula-solving computer constituted the only practical use of the involved algorithm, however, the Supreme Court deemed unpatentable what it considered claims to an algorithm performed in a computer, i. e., claims to the algorithm itself.
The precedent for the Benson “nutshell” language is found in two basic propositions. The first is that “[phenomena of nature, though just discovered, mental processes, and abstract intellectual concepts are not patentable, as they are the basic tools of scientific and technological work.” 409 U.S. at 67, 93 S.Ct. at 255, 175 USPQ at 675. Support for that proposition appears in MacKay Co. v. Radio Corp. of America, 306 U.S. 86, 59 S.Ct. 427, 83 L.Ed. 506 (1939); Rubber-Tip Pencil Co. v. Howard, 87 U.S. (20 Wall.) 498 22 L.Ed. 410 (1874); LeRoy v. Tatham, 55 U.S. 167, 14 How. 156, 14 L.Ed. 367 (1852); and Funk Bros. Seed Co. v. Kalo Inoculant Co., 333 U.S. 127, 68 S.Ct. 440, 92 L.Ed. 588 (1948). A corollary was stated in MacKay: “While a scientific truth, or the mathematical expression of it, is not a patentable invention, a novel and useful structure created with the aid of knowledge of scientific truth may be.”3 *1242306 U.S. at 94, 59 S.Ct. at 431. To “novel and useful structure,” encompassed within “machine,” should be added the other statutorily patentable categories, i. e., processes, manufacturers, compositions of matter, and new and useful improvements thereof.
The second proposition is that a statutory claim must define more than a mere effect. In O'Reilly v. Morse, 56 U.S. 65, 15 How. 62, 14 L.Ed. 601 (1853), cited by the Court in Benson, it was alleged that a Morse claim was void because drawn to “a principle or effect, and not a machine, manufacture, or composition of matter, or an improvement upon either.” 56 U.S. at 106, 15 How. at 100. Morse sought to claim:
Eighth. I do not propose to limit myself to the specific machinery or parts of machinery described in the forgoing specification and claims; the essence of my invention being the use of the motive power of the electric or galvanic current, which I call electro-magnetism, however developed for marking or printing intelligible characters, signs or letters, at any distances, being a new application of that power of which I claim to be the first inventor or discoverer. [56 U.S. at 119, 15 How. at 112.]
The Supreme Court characterized the claim in Morse:
It is impossible to misunderstand the extent of this claim. He claims the exclusive right to every improvement where the motive power is the electric or galvanic current, and the result is the marking or printing intelligible characters, signs, or letters at a distance.
If this claim can be maintained, it matters not by what process or machinery the result is accomplished. For aught that we now know some future inventor, in the onward march of science, may discover a mode of writing or printing at a distance by means of the electric or galvanic current, without using any part of the process or combination set forth in the plaintiff’s specification. His invention may be less complicated — less liable to get out of order — less expensive in construction, and in its operation. But yet if it is covered by this patent the inventor could not use it, nor the public have the benefit of it without the permission of this patentee. [56 U.S. at 119-20, 15 How. at 112-13.]
Though principally involving an issue raisable today under § 112, Morse illustrates the Supreme Court’s concern that an “effect” shall not be deemed patentable subject matter. A claimed process or machine for implementing an “effect,” however, is patentable, The Telephone Cases, 126 U.S. 1, 534, 8 S.Ct. 778, 31 L.Ed. 863 (1887), and that is so regardless of whether it constitutes the only known means of accomplishing the “effect”:
We see nothing in Morse’s case to defeat Bell’s claim; on the contrary, it is in all respects sustained by that authority. It may be that electricity cannot be used at all for the transmission of speech except in the way Bell has discovered, and that therefor, practically, his patent gives him its exclusive use for that purpose, but that does not make his claim one for the use of electricity distinct from the particular process with which it is connected in his patent. [126 U.S. at 535, 8 S.Ct. at 782, emphasis added.]
If such were not the law, the anomalous result would occur that an inventor who *1243first discovered a practical utility for a principle, law or force of nature, or their mathematical expression, could not obtain a patent on a particular process employing that practical utility. The inventor would have to wait until (if ever) he discovered at least one other practical utility, and the public would have to wait for a disclosure of the inventor’s initial discovery.4
The type of problem facing the Supreme Court in Morse, The Telephone Cases, and Benson, was described by one commentator over a hundred years ago, as “how far a discovery or invention which may first disclose and practically embody some truth in physics or some law in the operation of the forces of nature, for a useful purpose, is capable of being carried in the exclusive privileges secured by the grant of letters-patent,” Curtis, A Treatise on the Law of Patents For Useful Inventions § 124 (4th ed. 1873). Recent authority is in accord. See 1 Deller’s Walker on Patents § 23 (1964).
It is thus clear that the “nutshell” language of Benson, expressed the ancient rule that practical application remains the key. Because it did not consider the performance of an algorithm by a computer as constituting a practical application of that algorithm under the rule, the Court must have viewed Benson’s claims as effectively claiming the “effect,” principle, or law or force of nature (the algorithm) itself.
Claim Analysis
The distinction may thus be fine indeed between statutory and nonstatutory subject matter, considering the glorious flexibility and frustrating limitations of the English language on the one hand, and the ingenuity of patent draftsmen on the other. Nonetheless, the line required by precedent, and which must here be drawn, is clear. The mathematical expression of scientific truth or principle is itself not patentable. “The decisive factor is whether a claimed method is essentially a mathematical calculation.” In re Richman, 563 F.2d 1026, -USPQ-(Cust. & Pat.App.1977).
In In re Chatfield, supra, the claims were drawn to a method of operating a computer machine system. Though it employed solutions from equations, the method simply used the results of those equations, and the claims were not drawn to an equation or algorithm per se. In In re Deutsch, 553 F.2d 689, 193 USPQ 645 (Cust. & Pat.App. 1977), the claims were drawn to a method of operating a system of manufacturing plants, the method used the results of an algorithm, and the algorithm was not the method nor the method the algorithm. In In re Flook, 559 F.2d 21, - USPQ - (Cust. & Pat.App.1977), unlike the situation in In re Christensen, 478 F.2d 1392, 178 USPQ 35 (Cust. & Pat.App.1973), the post-algorithm solution activity recited in the claims established that the claimed method involved simply the use of an algorithm, and the claim was not in effect a claim to the algorithm per se.
On the other hand, in In re Waldbaum, 559 F.2d 611, 194 USPQ 465 (Cust. & Pat. App.1977), a number of the claims, drawn to an algorithm with no practical application other than in a data processing apparatus, were deemed nonstatutory under the tenets of Benson, and other claims, limiting the algorithm to a particular environment, were nonetheless directed solely to a process for calculating, i. e., to an algorithm per se. The same was true of the claims in In re Richman, supra.
De Castelet’s claim 7 begins by describing his method as one for “generating a curve from data supplied to a computer.” According to the claims, the ultimate objective, curve generation, is achieved through use of a computer instructed (programmed) to perform certain calculations upon stored and inputed data.
Specifically, the inputed data are “supplied to a computer in the form of coordinates of [end] points defining two given segments of tangents to the curve to be generated.” The data stored are “in the *1244form of electrical signals representing a table of coordinates of points, of characteristics of a base curve inscribed on a unit-cube between two opposite vertices of said unit-cube.” The computer is instructed “(1) to treat electrical signals representing a given arc, defined [by the inputed data] * * * as the transformed curve of said base curve, wherein [the inputed data] * * are considered as the transformed points of vertices of said unit-cube and (2) to calculate and transmit to the control system of said model forming means electrical signals representing the coordinates of a sequence of points * * * of said given arc defining said curve to be generated.”
The recited method steps are computation steps performed by the computer pursuant to instructions “(1)” and “(2).” Instruction “(2)” also includes the non-computation step of transmitting electrical signals to a “model forming means.”
Overall, therefore, de Castelet’s claimed method involves the storing of certain mathematical data in a computer, inputing additional mathematical data, causing the computer to perform programmed computations using the stored and inputed data, and, finally, causing the computer to transmit the results of those computations to a “model forming means.”
We think the instant claims recite a process for solving a set of mathematical equations per se, the solution being a set of points along a curve, and not a process which merely uses equation solutions as one step in achieving some result other than solution of the equations. We hold that de Castelet’s claims are drawn to nonstatutory subject matter.5
The claimed method is not one for operating a machine system or plant system in a particular manner, as in Chatfield, supra, or Deutsch, supra. The preamble recites that de Castelet’s claimed method is for generating a curve. The method steps claimed, however, are directed to computer instructions for solving mathematical equations and transmitting electrical signals representing those solutions.
That the computer is instructed to transmit electrical signals, representing the results of its calculations, does not constitute the type of “post solution activity” found in Flook, supra, and does not transform the claim into one for a process merely using an algorithm. The final transmitting step constitutes nothing more than reading out the result of the calculations. Recitations of specific machine elements, i. e., the mere reference in the claims to a computer and a model-former, cannot alone render statutory the presently claimed subject matter as a whole. Claims to nonstatutory processes do not automatically and invariably become patentable upon incorporation of reference to apparatus.6
De Castelet’s reliance on In re Bernhardt, 417 F.2d 1395, 57 CCPA 737, 163 USPQ 611 (1969) is misplaced. The basis for the PTO rejection in Bernhardt being unclear, we deemed it grounded on a “mental steps” theory:
In the case now before us, the disclosure shows only machinery for carrying out the portrayal process. In fact it is the chief object of the invention to eliminate the drudgery involved in a draftsman’s making the desired portrayals. Accordingly, a statutory process is here disclosed. Looking then to method claim 13, we find that it in no way covers any mental steps but requires both a “digital computer” and a “planar plotting apparatus” to carry it out. To find that the claimed process could be done mentally *1245would require us to hold that a human mind is a digital computer or its equivalent, and that a draftsman is a planar plotting apparatus or its equivalent. On the facts of this case we are unwilling so to hold. We conclude that the method defined by claim 13 is statutory, and its patentability must be judged in light of the prior art. [417 F.2d at 1401, 57 CCPA at 745, 163 USPQ at 617.]
We recognize that de Castelet’s claims include a computer and a drafting or machine tool and, accordingly, are not drawn to “mental steps,” but there is no “mental steps” rejection before us. Similarly, the basis of the present rejection was not before us in Bernhardt. That the same section of the statute is employed does not imply the same rationale underlying a rejection.
However, in Bernhardt this court cautioned against allowance of claims preempting equations, and what was there said, regarding § 101 in general, does have relevance here:
We think it is clear that in enacting section 101 Congress meant to exclude principles or laws of nature and mathematics, of which equations are an example, from even temporary monopolization by patent. [417 F.2d at 1399, 57 CCPA at 743, 163 USPQ at 616.]
The present equations remain “basic tools of scientific and technological work,” 409 U.S. at 67, 93 S.Ct. at 255,175 USPQ at 675, when resolved by a computer and when computer resolution produces corresponding electrical signals. In sum, we are convinced, as was the Supreme Court in Benson, that the resolution of de Castelet’s equations by a computer, which then transmits the electrical result to a tool, is not a practical application within the rule, and that a patent containing the appealed claims would in effect be no more than a patent on de Castelet’s equations.
Hence the present ease falls on that side of the statutory-nonstatutory line occupied, for example, by Morse, Benson, Christensen, Waldbaum, and Richman, and not on that occupied by The Telephone Cases, Chatfield, Deutsch, and Flook. Accordingly, the decision of the board is affirmed.
AFFIRMED.
RICH, and LANE, JJ., concur in result.
. 35 U.S.C. § 101 provides:
Whoever invents or discovers any new and useful process, machine, manufacture, or composition of matter, or any new and useful improvement thereof, may obtain a patent therefor, subject to the conditions and requirements of this title.
Broader questions regarding general patentability of computer programs or computer-implemented processes must await the action of Congress. Dealing only with the particular circumstances of the present appeal, we reiterate that the analysis under § 101 must be directed to the claimed subject matter as a whole. Chatfield, 545 F.2d at 158, 191 USPQ at 736. We specifically reject the solicitor’s suggestion that a piecemeal analysis is either useful or appropriate.
. The examiner rejected earlier claims under 35 U.S.C. § 103, citing substantial prior art, of which a listing appears in the record, and asserting that the art disclosed computer controlled drafting and machine tools and methods of generating continuous curves from predetermined data points. The record is silent with respect to the reasons for not applying the same rejection to claims 5-7. The application was passed to issue, containing claims 5-7, and withdrawn from issue after the Supreme Court’s decision in Benson.
. Thus, it is important to distinguish between situations in which (1) the scientific truth per se constitutes the sum of the subject matter sought to be patented, and (2) the scientific truth is merely used in an overall process or to describe a physical relationship.
In MacKay, RCA owned a patent claiming a particular antenna structure having a relation*1242ship of elements conforming to a mathematical formula:
15. An antenna comprising a pair of relatively long conductors disposed with respect to each other at an angle substantially equal to twice
degrees, 1 being the length of the wire and A the operating wave length in like units, ánd means in circuit with said antenna for exciting the conductors in phase opposition whereby standing waves of opposite instantaneous polarity are formed on the conductors throughout their length.
The formula per se would not have been patentable, but its use in describing the new and useful antenna device would not render a claim to the device unpatentable.
. The anomaly is similar to that observed in In re Tarczy-Hornoch, 397 F.2d 856, 55 CCPA 1441, 158 USPQ 141 (1968), attendant the “mere function of a machine” rejection.
. We voice no opinion on the patentability of disclosed but unclaimed subject matter.
. A recitation that a process is performed by a machine may preclude rejection on a “mental steps” theory. See In re Prater, 415 F.2d 1393, 56 CCPA 1381, 162 USPQ 541 (1969).
Though the preamble of the appealed claims refers to curve generation, the recited steps do not result in the formation of a curved drawing or surface. The “model forming means” is merely recited as receiving signals.
The potential for misconstruction of preamble language requires that compelling reason exist before that language may be given weight. No such reason has been presented in this case.